A History of Greek Mathematics and Astronomy

Page 5

The principles of geometry and arithmetic (in the sense of the theory of numbers) are stated in the preliminary matter of Books I and VII of Euclid. But Euclid was not their discoverer; they were gradually evolved from the time of Pythagoras onwards. Aristotle is clear about the nature of the principles and their classification. Every demonstrative science, he says, has to do with three things, the subject-matter, the things proved, and the things from which the proof starts (εξ ὡν {ex hôn}). It is not everything that can be proved, otherwise the chain of proof would be endless; you must begin somewhere, and you must start with things admitted but indemonstrable. These are, first, principles common to all sciences which are called axioms or common opinions, as that 'of two contradictories one must be true', or 'if equals be subtracted from equals, the remainders are equal'; secondly, principles peculiar to the subject-matter of the particular science, say geometry. First among the latter principles are definitions; there must be agreement as to what we mean by certain terms. But a definition asserts nothing about the existence or non-existence of the thing defined. The existence of the various things defined has to be proved except in the case of a few primary things in each science the existence of which is indemonstrable and must be assumed among the first principles of the science; thus in geometry we must assume the existence of points and lines, and in arithmetic of the unit. Lastly, we must assume certain other things which are less obvious and cannot be proved but yet have to be accepted; these are called postulates, because they make a demand on the faith of the learner. Euclid's Postulates are of this kind, especially that known as the parallel-postulate.

The methods of solution of problems were no doubt first applied in particular cases and then gradually systematized; the technical terms for them were probably invented later, after the methods themselves had become established.

One method of solution was the reduction of one problem to another. This was called απαγωγη {apagôgê}, a term which seems to occur first in Aristotle. But instances of such reduction occurred long before. Hippocrates of Chios reduced the problem of duplicating the cube to that of finding two mean proportionals in continued proportion between two straight lines, that is, he showed that, if the latter problem could be solved, the former was thereby solved also; and it is probable that there were still earlier cases in the Pythagorean geometry.

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T. L. Heath
A History of Greek Mathematics and Astronomy

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T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy